The video below from TED Ed shows a brief history of how languages evolve, as speakers of the same language lose contact with each other in the centuries after migration and gradually drift linguistically in different directions.

What's most interesting is not simply how we got multiple languages but rather how we determine, without the benefit of a time machine, which modern languages are related. To do this, historical linguists compare large numbers of words in different languages, looking for similarities that can't be explained by other factors, such as onomatopoeia (the word for cat is something like "miao" in several languages, but, well, there's likely an obvious reason for that) or borrowing (the word for tea in most languages is something like te or cha, but those can both be traced back to trade routes from different parts of China).